/ a
V
/
C 1
Figur e 1.10. Snell's law
Exercise 1.13. Deduce Snell's law from the Fermat p
To describe optical properties of the medium, o
"unit sphere" S(X) at every point X: it consists of th
vectors at X. The hypersurface S is called indicatrix;
smooth, centrally symmetric and strictly convex. For e
case of Euclidean space, the indicatrices at all points
unit spheres. A field of indicatrices determines the so
metric: the distance between points A and B is the lea
light to get from A to B. A particular case of Finsler g
Riemannian one. In the latter case, one has a (varia
structure in the tangent space at every point X, and
S(X) is the unit sphere in this Euclidean structure.
Another example is a Minkowski metric. This is a
in a vector space whose indicatrices at different point
from each other by parallel translations. The speed
Minkowski space depends on the direction but not th
a homogeneous but anisotropic medium. Minkowski's
the study of these geometries came from number theo
Propagation of light satisfies the Huygens principl
A and consider the locus of points Ft reached by light
t. The hypersurface Ft is called a wave front, and it
points at Finsler distance t from A. The Huygens p
that the front Ft+£ can be constructed as follows: ever
lrThere was a heated polemic between Fermat and Descartes c
the speed of light increases or decreases with the density of the m
erroneously thought that light moves faster in water than in the air
Previous Page Next Page